Non-Gaussianity and the Cosmic Microwave Background Anisotropies

TL;DR

This work analyzes how non-Gaussianity imprints in CMB anisotropies arise from both primordial origins and non-linear post-inflationary physics. It develops a comprehensive second-order cosmological perturbation framework, deriving gauge-invariant large-scale results and a full Boltzmann treatment at all scales, including the photon-baryon fluid, collision terms, and matter perturbations. The review quantifies secondary NG contamination to primordial NG via analytic estimates and Fisher-matrix analyses, and outlines detailed numerical schemes to compute the second-order CMB bispectrum, highlighting recombination-era contributions and their impact on equilateral versus local NG measurements. Overall, it provides a rigorous, multi-scale methodology to disentangle primordial NG from secondary sources, with implications for interpreting current and upcoming CMB data in terms of early-universe physics.

Abstract

We review in a pedagogical way the present status of the impact of non-Gaussianity (NG) on the Cosmic Microwave Background (CMB) anisotropies. We first show how to set the initial conditions at second-order for the (gauge invariant) CMB anisotropies when some primordial NG is present. However, there are many sources of NG in CMB anisotropies, beyond the primordial one, which can contaminate the primordial signal. We mainly focus on the NG generated from the post-inflationary evolution of the CMB anisotropies at second-order in perturbation theory at large and small angular scales, such as the ones generated at the recombination epoch. We show how to derive the equations to study the second-order CMB anisotropies and provide analytical computations to evaluate their contamination to primordial NG (complemented with numerical examples). We also offer a brief summary of other secondary effects. This review requires basic knowledge of the theory of cosmological perturbations at the linear level.

Figures (7)

• Figure 1: Shape dependence of the second-order bispectrum from products of the first-order terms (top) and that of the local primordial bispectrum (bottom). We show $\ell_1\ell_2\langle a_{\ell_1m_1}^{(1)}a_{\ell_2m_2}^{(1)}a_{\ell_3m_3}^{(2)}\rangle {({\cal{G}}_{\ell_1\ell_2\ell_3}^{m_1m_2m_3})}^{-1}/(2\pi)^2\times 10^{22}$ as a function of $\ell_1/\ell_3$ and $\ell_2/\ell_3$ where $\ell_3=200$. Both shapes have the largest signals in the squeezed triangles, $\ell_1\ll \ell_2\approx \ell_3$.
• Figure 2: Same as Fig. \ref{['fig:shape1']} for $\ell_3=1000$. The acoustic oscillations are clearly seen.
• Figure 3: Signal-to-noise ratios for the local primordial bispectrum for $f_{\rm NL}=1$ (dashed), and the second-order bispectrum from the products of the first-order terms (solid), for an ideal full-sky and cosmic-variance-limited (noiseless) experiment.
• Figure 4: Absolute values of the contributions to the signal-to-noise ratio from each component, $(S/N)_{ab}$, as defined by Eq. (\ref{['eq:sncomp']}).
• Figure 5: The cross-correlation coefficient between the second-order bispectrum from the products of the first-order terms and the local primordial bispectrum.